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Foster B, Knobloch E. Elastic fingering in a rotating Hele-Shaw cell. Phys Rev E 2023; 107:065104. [PMID: 37464645 DOI: 10.1103/physreve.107.065104] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2022] [Accepted: 06/05/2023] [Indexed: 07/20/2023]
Abstract
We consider the steady-state fingering instability of an elastic membrane separating two fluids of different density under external pressure in a rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are considered, and the role of membrane tension is detailed in each case. Both systems exhibit a centrifugally driven Rayleigh-Taylor-like instability when the density of the inner fluid exceeds that of the outer one, and this instability competes with the restoring forces arising from curvature and tension, thereby setting the finger scale. Numerical continuation is used to compute not only strongly nonlinear primary finger states up to the point of self-contact, but also secondary branches of mixed modes and circumferentially localized folds as a function of the rotation rate and the externally imposed pressure. Both reflection-symmetric and symmetry-broken chiral states are computed. The results are presented in the form of bifurcation diagrams. The ratio of system scale to the natural length scale is found to determine the ordering of the primary bifurcations from the unperturbed circle state as well as the solution profiles and onset of secondary bifurcations.
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Affiliation(s)
- Benjamin Foster
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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Foster B, Verschueren N, Knobloch E, Gordillo L. Universal Wrinkling of Supported Elastic Rings. PHYSICAL REVIEW LETTERS 2022; 129:164301. [PMID: 36306759 DOI: 10.1103/physrevlett.129.164301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Accepted: 09/19/2022] [Indexed: 06/16/2023]
Abstract
An exactly solvable family of models describing the wrinkling of substrate-supported inextensible elastic rings under compression is identified. The resulting wrinkle profiles are shown to be related to the buckled states of an unsupported ring and are therefore universal. Closed analytical expressions for the resulting universal shapes are provided, including the one-to-one relations between the pressure and tension at which these emerge. The analytical predictions agree with numerical continuation results to within numerical accuracy, for a large range of parameter values, up to the point of self-contact.
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Affiliation(s)
- Benjamin Foster
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Nicolás Verschueren
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
- College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF, United Kingdom
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Leonardo Gordillo
- Departamento de Física, Facultad de Ciencia, Universidad de Santiago de Chile, Estación Central 9170124, Chile
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Anjos PHA, Carvalho GD, Lira SA, Miranda JA. Wrinkling and folding patterns in a confined ferrofluid droplet with an elastic interface. Phys Rev E 2019; 99:022608. [PMID: 30934336 DOI: 10.1103/physreve.99.022608] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2018] [Indexed: 11/07/2022]
Abstract
A thin elastic membrane lying on a fluid substrate deviates from its flat geometry on lateral compression. The compressed membrane folds and wrinkles into many distinct morphologies. We study a magnetoelastic variant of such a problem where a viscous ferrofluid, surrounded by a nonmagnetic fluid, is subjected to a radial magnetic field in a Hele-Shaw cell. Elasticity comes into play when the fluids are brought into contact, and due to a chemical reaction, the interface separating them becomes a gel-like elastic layer. A perturbative linear stability theory is used to investigate how the combined action of magnetic and elastic forces can lead to the development of smooth, low-amplitude, sinusoidal wrinkles at the elastic interface. In addition, a nonperturbative vortex sheet approach is employed to examine the emergence of highly nonlinear, magnetically driven, wrinkling and folding equilibrium shape structures. A connection between the magnetoelastic shape solutions induced by a radial magnetic field and those produced by nonmagnetic means through centrifugal forces is also discussed.
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Affiliation(s)
- Pedro H A Anjos
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - Gabriel D Carvalho
- Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, Rio de Janeiro 22290-180, Brazil
| | - Sérgio A Lira
- Instituto de Física, Universidade Federal de Alagoas, Maceió, Alagoas 57072-900, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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Gordillo L, Knobloch E. Fluid-supported elastic sheet under compression: Multifold solutions. Phys Rev E 2019; 99:043001. [PMID: 31108605 DOI: 10.1103/physreve.99.043001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2018] [Indexed: 11/07/2022]
Abstract
The properties of a hinged floating elastic sheet of finite length under compression are considered. Numerical continuation is used to compute spatially localized buckled states with many spatially localized folds. Both symmetric and antisymmetric states are computed and the corresponding bifurcation diagrams determined. Weakly nonlinear analysis is used to analyze the transition from periodic wrinkles to singlefold and multifold states and to compute their energy. States with the same number of folds have energies that barely differ from each other and the energy gap decreases exponentially as localization increases. The stability of the different competing states is studied and the multifold solutions are all found to be unstable. However, the decay time into solutions with fewer folds can be so slow that multifolds may appear to be stable.
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Affiliation(s)
- Leonardo Gordillo
- Departamento de Física, Universidad de Santiago de Chile, Av. Ecuador 3493, Estación Central, Santiago, Chile
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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Fontana JV, Miranda JA. Elastic fingering patterns in confined lifting flows. Phys Rev E 2016; 94:033110. [PMID: 27739751 DOI: 10.1103/physreve.94.033110] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2016] [Indexed: 11/07/2022]
Abstract
The elastic fingering phenomenon occurs when two confined fluids are brought into contact, and due to a chemical reaction, the interface separating them becomes elastic. We study elastic fingering pattern formation in Newtonian fluids flowing in a lifting (time-dependent gap) Hele-Shaw cell. Using a mode-coupling approach, nonlinear effects induced by the interplay between viscous and elastic forces are investigated and the weakly nonlinear behavior of the fluid-fluid interfacial patterns is analyzed. Our results indicate that the existence of the elastic interface allows the development of unexpected morphological behaviors in such Newtonian fluid flow systems. More specifically, we show that depending on the values of the governing physical parameters, the observed elastic fingering structures are characterized by the occurrence of either finger tip splitting or side branching. The impact of the elastic interface on finger-competition events is also discussed.
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Affiliation(s)
- João V Fontana
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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Fontana JV, Gadêlha H, Miranda JA. Development of tip-splitting and side-branching patterns in elastic fingering. Phys Rev E 2016; 93:033126. [PMID: 27078466 DOI: 10.1103/physreve.93.033126] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2016] [Indexed: 11/07/2022]
Abstract
Elastic fingering supplements the already interesting features of the traditional viscous fingering phenomena in Hele-Shaw cells with the consideration that the two-fluid separating boundary behaves like an elastic membrane. Sophisticated numerical simulations have shown that under maximum viscosity contrast the resulting patterned shapes can exhibit either finger tip-splitting or side-branching events. In this work, we employ a perturbative mode-coupling scheme to get important insights into the onset of these pattern formation processes. This is done at lowest nonlinear order and by considering the interplay of just three specific Fourier modes: a fundamental mode n and its harmonics 2n and 3n. Our approach further allows the construction of a morphology diagram for the system in a wide range of the parameter space without requiring expensive numerical simulations. The emerging interfacial patterns are conveniently described in terms of only two dimensionless controlling quantities: the rigidity fraction C and a parameter Γ that measures the relative strength between elastic and viscous effects. Visualization of the rigidity field for the various pattern-forming structures supports the idea of an elastic weakening mechanism that facilitates finger growth in regions of reduced interfacial bending rigidity.
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Affiliation(s)
- João V Fontana
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
| | - Hermes Gadêlha
- Department of Mathematics, University of York, York YO10 SDD, United Kingdom
| | - José A Miranda
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil
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Hota TK, Pramanik S, Mishra M. Nonmodal linear stability analysis of miscible viscous fingering in porous media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:053007. [PMID: 26651779 DOI: 10.1103/physreve.92.053007] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2015] [Indexed: 06/05/2023]
Abstract
The nonmodal linear stability of miscible viscous fingering in a two-dimensional homogeneous porous medium has been investigated. The linearized perturbed equations for Darcy's law coupled with a convection-diffusion equation is discretized using a finite difference method. The resultant initial value problem is solved by a fourth-order Runge-Kutta method, followed by a singular value decomposition of the propagator matrix. Particular attention is given to the transient behavior rather than the long-time behavior of eigenmodes predicted by the traditional modal analysis. The transient behaviors of the response to external excitations and the response to initial conditions are studied by examining the ε-pseudospectra structures and the largest energy growth function, respectively. With the help of nonmodal stability analysis we demonstrate that at early times the displacement flow is dominated by diffusion and the perturbations decay. At later times, when convection dominates diffusion, perturbations grow. Furthermore, we show that the dominant perturbation that experiences the maximum amplification within the linear regime lead to the transient growth. These two important features were previously unattainable in the existing linear stability methods for miscible viscous fingering. To explore the relevance of the optimal perturbation obtained from nonmodal analysis, we performed direct numerical simulations using a highly accurate pseudospectral method. Furthermore, a comparison of the present stability analysis with existing modal and initial value approach is also presented. It is shown that the nonmodal stability results are in better agreement than the other existing stability analyses, with those obtained from direct numerical simulations.
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Affiliation(s)
- Tapan Kumar Hota
- Department of Mathematics, Indian Institute of Technology Ropar, Nangal Road, 140001 Rupnagar, India
| | - Satyajit Pramanik
- Department of Mathematics, Indian Institute of Technology Ropar, Nangal Road, 140001 Rupnagar, India
| | - Manoranjan Mishra
- Department of Mathematics, Indian Institute of Technology Ropar, Nangal Road, 140001 Rupnagar, India
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